Systems affect signals by inducing both linear and nonlinear distortions, including those due to memory effects. Memory in this context means that a system response at a time instant of interest (that was otherwise intended to be uncorrelated with other time instances) depends on, and is influenced by, signal values in surrounding time instants. Some examples of electrical systems containing signals that suffer from linear and nonlinear distortion, including memory effects, include optical communication systems, wireless communication systems, satellite communication systems, computer data storage systems, image processing systems, video processing systems, sound processing systems, and controls systems, among others. The impairments (both linear and nonlinear) diminishing the quality of the useful signal are in practice commonly countered by some means of filtering, equalization, or compensation (often used interchangeably).
One method to compensate signals that have linear and nonlinear distortions with memory effects, is accomplished by filters that are based on polynomial expansions, where the filter output (y[n]) is a weighted linear combination of the input signal (x[n]) at different time instances, plus the input signal at different time instances raised to a given power (determined by the nonlinearity order), plus the (linear) combination of various cross-products of the input signal at different time instances each raised to an appropriate power. For example, filters with the full-complexity polynomial expansion based on the Volterra series are accordingly called Volterra filters. For example, the output of a Volterra filter of order 2 and memory of 2 will have the form:y[3]=c1·x[1]+c2·x[2]+c3·x[3]+c11·x[1]2+c22·x[2]2+c33·x[3]2+c11·x[1]·x[2]+c13·x[1]·x[3]+c23·x[2]·x[3]  (1)In Equation 1, ci are parameters multiplying the signal at the ith instance in time, cjj are parameters multiplying the squared term of the signal at the jth instance in time, and cjk (where j does not equal k) are parameters multiplying the signal cross terms at the jth and kth instances in time. However, filters employing the Volterra series are complex to implement in practice. Specifically, the number of parameters in the Volterra approach scales exponentially with the number of signal instances in surrounding time epochs that affect a given signal state (i.e., scales exponentially with memory). The exponential nature of nonlinear filters employing the Volterra series can be expensive to implement, and impractical in high-speed systems, due to the large amount of operations that are necessary to be performed, which in practice correspond to large amounts of RAM, or taps, or excessive power consumption associated with the number of computations required. Furthermore, filters employing the Volterra series rely on polynomial functions, and are not well equipped to mitigate distortions that, in practice, are not necessarily well approximated by continuous polynomials. For example, ADC and DAC components often introduce nonlinear distortions that are not well approximated by continuous polynomial functions, and are better approximated by piecewise continuous functions. As a result, systems with nonlinear signal distortions resulting from ADC or DAC components in combination with components that introduce memory effects in the signal are not well mitigated by filters employing the Volterra series.
Other methods for filtering signals that suffer from both linear and nonlinear distortion are accomplished using polynomial expansions including some, but not all, of the terms of the Volterra series. One such variation of Volterra filters are Memory Polynomial filters, which contain a portion of the terms in the Volterra series (e.g., only the diagonal terms) and do not contain the cross-products of the input signal at different time instances. For example, the first output (y[1]) of a Memory Polynomial filter of order 2 and memory of 3 will have the form:y[1]=c1·x[1]+c2·x[2]+c3·x[3]+c11·x[1]2+c22·x[2]2+c33·x[3]2  (2)In Equation 2, x[n], y[n], and the parameters, ci and cjj, are defined the same as they are in Equation 1. Nevertheless, even though memory polynomials are less complex than Volterra series, the price to pay for the omission of the cross-terms is a significant decline in performance, and they are equally limited to continuous polynomial forms.
Electronic communications systems typically convert a digital input signal into an analog form by upconverting, filtering, and amplifying the signal for transmission using analog components. The digital and analog components can achieve only limited accuracy, and nonlinear distortions with memory effects are commonplace. An example of components that can induce nonlinear distortions, in addition to the digitizers in communications systems are analog linear power amplifiers. As linear power amplifiers approach the end of their dynamic range, saturation can occur, which induces a departure from a linear behavior, or response, thus, if left unattended, leading to distortions, or otherwise departures from the intended signal shaping. Linear power amplifiers are some of the more expensive components in transmitters used in communications systems, and less expensive amplifiers tend to have worse nonlinearity. Additionally, high baud rate digital transmission systems tend to suffer from memory effects, which manifest as intersymbol interference (ISI). ISI is exacerbated in systems utilizing narrow band channels, such as telephone voice communication channels, where the channel response to one symbol is not allowed to transgress into the time-slot occupied by the next successive symbol.
High-speed information transmission in fiber-optic communication systems is another example of a system with linear and nonlinear distortions, including memory effects. In such systems, optical waves are used as carrier signals, and the information to be transmitted typically originates in electronic form as digital data. Prior to transmission, the electronic information is imprinted (e.g. modulated) onto an optical carrier signal by an optical transmitter. The modulated optical carrier signal is then transmitted over a fiber-optic, and is received by an optical receiver. Information from the received optical signal is then transformed back into an electronic form, such as digital data.
Imprinting electronic information onto an optical carrier signal can be performed by an electro-optical modulator, such as a dual-polarization Mach-Zehnder modulator (DP-MZM). A DP-MZM is capable of modulating information onto each of two orthogonal polarizations of a dual-polarized optical carrier signal.
In optical communications systems, as one example, memory effects can be caused by the frequency response of waveguides and electrodes within the DP-MZM in addition to the walk-off effect between the optical and electrical waves. Thus, impairments such as amplitude loss of the signal at a time t=0 can depend on a signal that was previously transmitted at time t=−1, can additionally depend on a signal transmitted at time t=−2, and so on. This can result in loss, change, or other impairments to information even before transmission (e.g., directly at the output of the DP-MZM). In addition to the memory effects, the MZM structure (and therefore, the DP-MZM, too) inherently possesses a nonlinear amplitude characteristic.
Another example of a type of optical communications system that can be affected by linear and nonlinear distortions with memory effects are those that utilize intensity modulated direct detection (i.e., IMDD systems). In IMDD systems, the power output of an optical source is modulated to encode a signal. The optical signal is transmitted through a channel (e.g., through free-space (i.e., air), or through an optical fiber), and then the signal is then received by a receiver. In an IMDD system, the receiver utilizes direct detection, for example by detecting the signal intensity using a photodiode. Such systems can operate in the wavelength band of visible or infrared light. An additional source of nonlinear distortions in IMDD systems is related to the fact that the power of the transmitted signal is related to the square of the intensity of the signal, and it is typically the signal intensity that is detected (e.g., by a photodiode) rather than the amplitude of the electric field, often referred to as the ‘square-law’ detection. Thus, the square-law detection acts in IMDD systems acts in addition to any further sources of nonlinearity such as MZIs, driver amplifiers, etc, further complicating detection in these systems.
Another example of a type of optical communications system that can be affected by linear and nonlinear distortions with memory effects are those that utilize heterodyning. Such systems encode information in an optical signal as modulation of the phase and/or frequency (i.e., wavelength) of electromagnetic radiation. A transmitter can encode heterodyne modulated information on an optical signal that is transmitted over a channel, and then detected in a receiver. Heterodyne detection in the receiver can employ local oscillators, and the detected signal can be compared with the reference light from the local oscillator (LO). Such systems can also operate in the wavelength band of visible or infrared light.
Some solutions attempt to reduce, mitigate or eliminate the effects of nonlinearity in electrical and optical transmission systems. Unfortunately, the presence of nonlinearities within circuitry of electrical and optical transmitters can reduce the ability to correct impairments at the receiver. Such attempted solutions often act on memoryless nonlinearity, and can thus be incapable of correcting impairments to the transmission signal that result from system memory effects.
Other solutions attempt to correct transmission signal impairments by using frequency response equalization at the transmitter. However, these solutions are often unsatisfactory because, in addition to the frequency response (e.g., the linear response of the system), transmitters often possess a nonlinear nature of the response that is inherent in the amplitude cosine transfer function of the transmitter.
Still other solutions seek to combine linear equalization, as well as equalizing the nonlinear response. However, these solutions are unsatisfactory because they neglect the system memory of the nonlinear response of the optical transmitters.
A common solution to correct for memory effects in communication systems are decision feedback equalizers (DFEs). DFEs typically employ a tapped delay line, which allows the equalizer to correct a signal suffering from memory effects, such as intersymbol interference. One type of DFE is a time domain DFE (TDDFE), where operations are performed on a symbol-by-symbol basis in the time domain. Several types of DFEs operate in the frequency domain, such as hybrid DFE (HDFE), extensionless HDFE (ELDFE), and iterative block DFE (IBDFE), which can reduce the computational complexity by roughly 25% compared with TDDFE. All DFEs can in principle compensate for memory effects in signals with nonlinearity, however, their effectiveness in mitigating nonlinear impairments is very limited and cannot adequately satisfy the requirements in modern communication, or control systems.